$\mathbb{Z}$-local system cohomology of hyperplane arrangements and a Cohen-Dimca-Orlik type theorem

TL;DR

This paper extends the Cohen-Dimca-Orlik vanishing theorem to $bZ$-local systems on hyperplane arrangement complements, enhancing understanding of their cohomology and topological properties.

Contribution

It proves a Cohen-Dimca-Orlik type theorem for $bZ$-local systems, generalizing previous results from $bC$-local systems.

Findings

Establishment of a vanishing theorem for $bZ$-local system cohomology

Generalization of known results from complex to integer local systems

Implications for topology and hypergeometric integrals

Abstract

Local system cohomology groups of the complements of hyperplane arrangements have played an important role in the theory of hypergeometric integrals, the topology of Milnor fibers and covering spaces. One of the important theorems is the vanishing theorem for generic $\mathbb{C}$-local systems which goes back to Aomoto's work. Later, Cohen, Dimca, and Orlik proved a stronger version of the vanishing theorem. In this paper, we prove a Cohen-Dimca-Orlik type theorem for $\mathbb{Z}$-local systems.